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A Sub Period Analysis of Long Memory in Stock Return Volatility
Shamila A. Jayasuriya*
* Assistant Professor, Department of Economics, Ohio University, Athens, OH 45701, USA.
Contact information: phone: +1 740-593-2094, fax: +1 740-593-0181, email: jayasuri@ohio.edu.
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ABSTRACT
Efficient market theory suggests that stock returns should not exhibit persistence or it would be
possible to generate trading profits by observing historical patterns. In this paper, we examine
the long run persistence of stock return volatility for 23 developing markets for the period
January 2000 to October 2007. The empirical analysis also includes a sub period investigation of
long memory and structural changes in volatility.
A FIEGARCH model is used for all
estimations. Results indicate persistence in return volatility for many markets. In addition, there
is no clear evidence that long memory can be attributed to structural changes in volatility.
JEL Classification: G14; G15
Keywords: Long memory; Stock return volatility; Emerging markets; FIEGARCH; Structural
changes
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INTRODUCTION
Long run persistence or long memory in stock return volatility has important implications
for the ability to predict future volatility and, therefore, for investment decisions that pertain to
the effective allocation of resources. A study of long memory, therefore, should be of interest to
investors and other agents who closely examine world equity markets with the aim of generating
trading profits. Stock indices that have useful information embedded in the past behavior of the
level and/or volatility of returns are essential to formulating future predictions of stock market
behavior that could result is substantial gains for investors. In recent years, developing markets
in general and emerging markets in particular have been the subject of close scrutiny by many
looking to diversify their portfolios. Developing markets have increasingly attracted the interest
of many foreign investors not only because of the relatively higher returns that they offer albeit
higher volatility but also because of the low correlations with developed markets that lead to
better diversification benefits. Therefore, a study of long memory that indicates predictability in
the long horizon for key developing markets of the world should be of use to many investors and
financial practitioners.
In this paper, we examine long memory in volatility for a large group 23 developing
markets. There is in fact much evidence of long-term dependence in stock return volatility for
many equity markets of the world as we will discuss in section 2 of the paper. Our paper
contributes to the existing literature in three main ways. First, we focus on a large group of
developing markets that include emerging and frontier markets from different regions of the
world. Second, we implement a sub period analysis for all the markets that examines how
consistent the long memory property has been over time. A final contribution is that we also
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identify structural breaks in the volatility series and investigate a link between long memory and
structural changes in volatility in a multi-country set up.
The empirical analysis is conducted for the time period from January 3, 2000 to October
15, 2007. A fractionally integrated exponential GARCH model that accounts for asymmetric
volatility is used for the empirical estimations. There is strong evidence of long run persistence
in volatility for all the markets although long memory is not observed for many in the most
recent sub period. This finding implies that many of the developing markets of the sample
appear to be efficient in terms of return volatility in the recent years. Using a robust outlier
detection procedure, we also identify structural changes in the volatility series. Subsequently we
find no conclusive evidence of a link between long memory and structural changes in volatility
even though some, especially a few of the emerging markets in Asia, appear to demonstrate such
a link.
The remainder of the paper is organized as follows. Section 2 discusses the literature
review. Section 3 describes the data and estimation methodology. Section 4 presents the
estimation results. And Section 5 concludes.
LITERATURE REVIEW
Many studies have empirically examined the long memory property of stock return
volatility.1 One group of studies applies long memory tests to various proxies for the return
volatility series such as the squared, log-squared, modified log-squared, and absolute returns.
Long memory tests that have been frequently used in the literature include the classical
periodogram based estimator of Geweke and Porter-Hudak (1983), the modified rescaled range
(R/S) statistic of Lo (1991), the rescaled variance (V/S) statistic of Giraitis et al. (2003), and the
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robust semiparametric procedure of Lobato and Robinson (1998). Another group of studies
applies the Fractional Integrated Generalized Autoregressive Conditional Heteroskedastic
(FIGARCH) model of Baillie, Bollerslev, and Mikkelsen (1996) to stock return data and tests for
the significance of a long memory parameter d in the conditional variance equation. Bollerslev
and Mikkelsen (1996) have later extended the FIGARCH model to the Fractionally Integrated
Exponential GARCH (FIEGARCH) process, which incorporates asymmetric shocks to the
model. A few studies fit into neither category and long memory in volatility is examined using
alternative models such as a SEMIFAR or Long Memory Stochastic Volatility (LMSV) models
instead. 2
Assaf (2004, 2006, 2007), Assaf and Cavalcante (2005), Chung et al (2000), Kilic (2004),
Sibbertsen (2004), So (2000), and Wright (2002) use one or more of the long memory tests in
their work. They test for long memory in return volatility using high frequency daily data
mainly for aggregate stock indices for both developed and less developed markets. 3
The
developed market indices that have been studied are the U.S. S&P 500 index, the Dow Jones
Industrial Average (DJIA) index, the Japanese Nikkei index, and an aggregate stock index for
Australia. Some of the developing or, in other words, emerging market indices that have been
studied include the aggregate indices for Brazil, Egypt, Jordan, Korea, Kuwait, Mexico,
Morocco, Taiwan, Thailand, and Turkey. All studies consistently find clear evidence of long
memory in return volatility. Results appear to be sensitive to the choice of volatility measure
with stronger evidence obtained for absolute than squared returns.4
Several authors raise the issue of whether the long memory effect is spurious or real by
detecting periods of volatility shifts and implementing long memory tests for each sub period.
Based on sub sample estimates, Assaf (2004, 2007) finds that long memory appears to be real
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and not due to structural shifts in the variance for several stock markets of the Middle East and
Africa (MENA) region. On the contrary, Chung et al (2000) find empirical evidence that support
spurious long memory due to shifts in variance for a group of seven Asia-Pacific markets.5
Similar to the immediately related work discussed above, other papers in the literature
have not been able to provide conclusive evidence on the link between long memory and
structural breaks as well. Diebold and Inoue (2001) use theoretical and simulation analyses to
argue that structural change in general, and stochastic regime switching in particular, are closely
related if only a small amount of structural change occurs in the sample. Granger and Hyung
(2004), too, present arguments based on theory and simulation results that show the difficulty in
distinguishing long memory in the occasional-break model and the I(d) model that is widely used
for series with long memory. In particular, the authors show that the volatility series may
indicate long memory because of the presence of neglected breaks. Choi and Zivot (2007) study
long memory and structural changes in the G7 countries’ forward discount. These authors find
clear evidence of long memory when structural breaks are not allowed in the forward discount
data. However, they also find evidence of stationary long memory even after adjusting for
multiple breaks in the data leading to the conclusion that the property of long memory is not
entirely due to structural breaks in the data. In a working paper, Hsu and Kuan (2000) propose
an econometric method called the local Whittle method that could jointly estimate the structural
break point and the long memory parameter. In an empirical application of this method, the
authors study monthly inflation rates in the G7 countries and find that the long memory effect on
inflation is robust to a one time structural break.
However, inflation persistence may be
overestimated if multiple breaks are not accounted for in the model.
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Assaf (2004, 2006), Assaf and Cavalcante (2005), Bellalah et al (2005), Kilic (2004), and
Wright (2002) estimate a FIGARCH model to examine long memory in volatility. With this type
of estimation, it is not necessary to proxy the volatility series of returns. Instead, volatility is
modeled from the data itself and the focus is on the significance and the magnitude of the
fractional integration parameter d. The above mentioned studies use daily or weekly data and are
based on the aggregate stock market indices for Egypt, Brazil, Kuwait, Tunisia, Turkey, and the
U.S., respectively. In all cases, the FIGARCH estimations produce a long memory parameter
that is highly significant and different from both zero and one, which indicate long memory in
the volatility series. In an extension to the FIGARCH model, Bollerslev and Mikkelsen (1996)
formulate the FIEGARCH model and apply it to daily returns of the U.S. S&P 500 index. They
find strong evidence that the conditional variance for the S&P 500 index is described well as a
mean-reverting fractionally integrated process. Also, a recent study by Saadi et al (2006) finds
that the FIEGARCH model provides the optimal fit for daily returns of the Tunisian stock market
because it captures high volatility persistence and long memory in the volatility of returns. We,
too, use a FIEGARCH model for all the estimations and document the existence of volatility
persistence for many emerging stock markets in our sample.
DATA AND ESTIMATION METHODOLOGY
The familiar GARCH(p,q) model is known to capture the volatility dynamics of stock
return data well. Particularly, it allows for volatility clustering that stock returns are known to
exhibit. In addition, the use of relevant autoregressive and moving average (ARMA) terms in
the mean specification allows for short run persistence in the returns series. The fractionally
integrated exponential GARCH (FIEGARCH) model extends the simple GARCH model to also
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account for asymmetric volatility effects and the long run persistence of volatility. Modeling an
exponential GARCH also guarantees that the conditional volatility of returns is always positive,
which often eliminates the need to impose certain parameter restrictions on the model
coefficients in order for stationarity to be achieved.
The general specification for the FIEGARCH(m,d,q) model that we estimate is given in
equations (1)–(4). Rt is the stock index return for a given emerging market, which has a
conditional distribution with mean t and variance  t2 .
 t ~ N (0,  t2 ) , and  t is an i.i.d.
sequence with zero mean and unit variance. For all estimations, we choose a parsimonious
model where we set m=q=1.6
Rt   t   t
(1)
a
b
i 1
j 1
 t     i Rt i   j  t  j
(2)
 t   t t
(3)
q
 ( L)(1  L) d ln  t2      i |  t i |  i t i 
(4)
i 1
where  ( L)  1  1 L   2 L2  ...   m Lm .
The  estimates measure volatility clustering or GARCH effects in the data with positive values
implying that higher (lower) volatility of stock returns in the past are followed by higher (lower)
volatility today. In addition, the  estimates capture the ARCH effects or the impact of past
news about volatility on current volatility. Also, asymmetric volatility or leverage of returns is
modeled by the  coefficients with negative values indicating that negative shocks have a bigger
impact on volatility than positive shocks of the same magnitude. For purposes of our paper, the
coefficient estimate of primary interest is the fractional difference parameter d that models the
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long run persistence of volatility. As Bollerslev and Mikkelsen (1996) show, the FIEGARCH
specification is stationary if 0 < d < 1.
The existence of long memory is confirmed for
significant values of d. For d = 0 and   0 , the above FIEGARCH model in fact reduces to the
familiar EGARCH model of Nelson (1991).
We study a large group of 24 markets that belong to different regions of the world. Most
are emerging markets in Asia, Europe, Middle East and North Africa (MENA), and Latin
America. A few are frontier markets, which are considered to be developing markets growing in
size and sophistication although they have not yet achieved the emerging market status. One
developed market, the U.S., is also examined for purpose of comparison. See Table 1 for a list
of the countries studied. The choice of markets is driven by our interest to study as many
developing markets as possible from different regions of the world. The list of countries on
Table 1 is limited to 24 primarily because of data availability at the daily frequency.
In
particular, daily data for the stock indices are collected from the Datastream database for the time
period from January 3, 2000 to October 15, 2007.7 Prices are in local currency and returns are
measured as one hundred times the log difference of stock price.
Table 1 also documents the basic summary statistics for all the markets. 8 As can be seen,
the average daily returns are generally higher for the emerging and frontier markets compared
with the U.S. market. The standard deviation of returns also appears to be generally higher for
the developing markets.
In addition, the stock returns for all markets are skewed and
consistently leptokurtic and are unlikely to have been drawn from a normal distribution. Table 2
provides relevant test statistics and their p-values that examine short memory and ARCH effects
of stock index returns. For example, the Ljung-Box Q-statistics at lags 12 and 24 formally test
the null hypothesis of no serial correlation in the returns series at the selected lags. Based on the
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p-values, we reject the null and observe the need to add autoregressive and moving average
terms in the conditional mean specification for many of the returns series. Furthermore, the
ARCH Lagrange Multiplier tests confirm volatility clustering for all the returns series justifying
the use of a variant of GARCH model as a good choice of estimation technique.
A preliminary exercise for evidence of long memory in volatility is to test for serial
correlation in the volatility series at a very long lag level. We select the squared and absolute
returns series as good proxies for return volatility and test for serial correlation at a lag level of
100.9 See Table 3 for results. Based on the squared returns, the relevant p-values of the LjungBox Q-statistics indicate that the null hypothesis of no serial correlation can be rejected at
conventional significance levels for all volatility series. In other words, there is preliminary
evidence that stock return volatility is predictable over the long horizon for the sample of
markets studied.
Based on the absolute returns, too, there is initial evidence of long run
persistence of volatility for all the indices.
In the next section, we will present formal evidence of long memory based on the
fractional parameter d of the FIEGARCH estimations. In addition, a sub period analysis of the
estimations will examine whether long memory in return volatility is consistent for a given
market or whether it depends on the time period studied. As a final exercise, we also implement
a robust outlier detection procedure to identify structural breaks and level shifts in the volatility
series. This may provide evidence of a link between long memory and structural changes in
volatility, which is an issue that has been studied by others in the existing literature but one that
has not given a clear consensus on the findings.
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EMPIRICAL RESULTS
FIEGARCH Estimation Results
Table 4 presents the FIEGARCH estimation results for the entire sample. We observe
reasonable and highly significant GARCH and ARCH effects for the different markets. Similar
to the U.S. market, the GARCH effects or volatility clustering is more prominent than ARCH
effects for many developing markets with the exception of Chile, Egypt, Morocco, and Tunisia.
The emerging markets of Asia and Latin America are characterized by asymmetric volatility as
indicated by significant negative leverage estimates although the magnitude of asymmetry is not
as high as in the U.S. A few emerging markets in the EMEA region such as Egypt, Jordan,
Morocco, Poland, and Slovakia and the two frontier markets of Lithuania and Tunisia do not
indicate asymmetric volatility effects after all. That is, for these markets, negative and positive
shocks of the same magnitude have a similar impact on return volatility. Interestingly, the
fractional difference parameter is highly significant and lies between a value of zero and one for
all stock index returns. Therefore, there is clear evidence of long memory in volatility for the
sample of countries investigated. That is, there is useful information embedded in the past stock
return volatility series that can be utilized for future predictions. The finding that markets do
have long memory in return volatility implies that investors may gain unrealized trading profits
by observing past behavior of return indices.
Table 5 documents the sub period FIEGARCH estimates for the leverage and fraction
terms.10 We have selected two sub periods of approximately equal length for our analysis. The
first sub period from January 3, 2000 to December 31, 2003 includes the September 11 attacks
on the United States. The second sub period from January 1, 2004 to October 15, 2007 does not
particularly include any major shocks to world equity markets. At the sub period level, too, we
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continued to observe reasonable and significant GARCH and ARCH effects for the markets. As
Table 5 documents, asymmetric volatility is not always present in the data. In fact, asymmetry is
not observed for both sub periods for Jordan, Lithuania, Morocco, Poland, Slovakia, and Tunisia.
For some markets including Chile, China, Egypt Malaysia, and Peru, asymmetric volatility is
observed only for one of the two sub periods. In addition, the magnitude of asymmetry for the
U.S. market is the highest for sub period one, and it is also relatively higher than most emerging
and frontier markets for sub period two.
A vast literature on the asymmetric volatility of returns alludes to the fact that there is no
one explanation for asymmetry. Some studies including Sentana and Wadhwani (1992) and
McQueen and Vorkink (2004) present explanations of asymmetric volatility based on models of
heterogeneous traders and behavioral finance. Such studies have considered the interaction
between traders and the arrival of news in the context of modern trading practices, especially
prevalent in developed and the more advanced emerging markets, which could contribute to
asymmetric volatility. A recent study by Jayasuriya, Shambora and Rossiter (2009) demonstrate
asymmetric volatility for several mature and emerging markets and suggest that asymmetry may
be linked to trading costs and trading strategies such as short selling. Furthermore, Brooks
(2007) investigates asymmetric volatility for a group of 26 emerging markets and reports high
asymmetry for markets in Latin America and low asymmetry for markets in Africa and the
Middle East. He also provides evidence that this observed variation in asymmetry cannot be
explained by differences in market size, thin trading or anti-director rights.
Lastly, we observe the fractional integration parameter estimates for the two sub periods.
For the U.S. market, there is significant evidence of long memory in volatility for the first but not
the second sub period. Many of the emerging and frontier markets also follow a similar pattern
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based on conventional significance levels.
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For example, the coefficient estimate for the
fractional difference parameter d is not significant and near zero in magnitude for many of the
developing markets for sub period two. This implies the non-existence of long memory during
the latter half of the time period for many of the markets studied. For Ecuador, Morocco,
Poland, and Slovakia, long memory in volatility exists for the second and not the first sub period.
For Israel, on the other hand, there is no evidence of long memory for both sub periods.11
Recall that our earlier results clearly indicated long memory in volatility when we
examined stock returns for the entire time period. Subsequently the sub period analysis suggests
that the long run persistence of shocks to volatility mainly does not exist in the most recent past,
which implies possible market efficiency at least in terms of return volatility in the recent years.
An intuitive explanation is that, as emerging markets continue to develop and grow, there would
be greater market participation that would in fact result in greater efficiency. For example, there
would be increased efficiency if a large number of competing, profit-maximizing participants
analyze the arrival of new information. In such a setting, security prices will adjust rapidly to the
release of all public information so that current prices fully reflect all available information and
there is no useful information left for future predictions. Also, with a large number of market
participants, no one group of investors typically has monopolistic access to information that is
used to determine equity prices. Relevant information, therefore, will be cost free and available
to everyone at the same time. Moreover, a higher level of stock market development is often
associated with greater market transparency, better accounting standards, and improved investor
protection laws. Such conditions would undoubtedly contribute to increased market efficiency.
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Structural Changes in Conditional Volatility
As discussed in section 2, several authors in the existing literature raise the possibility
that the observed long memory effect may be due to structural breaks and level shifts in the
return volatility series. The sub period analysis above is in fact an appropriate setting to identify
a link, if any, between the long memory property and structural changes in volatility. For
example, suppose a given market is observed to have long memory in volatility for sub period
one but not sub period two. We could then investigate whether sub period one experienced
significant structural breaks and shifts in the volatility series relative to sub period two.
For purposes of this study, the relevant volatility series is the fractionally integrated
conditional variance of returns shown in equation (4).
In Figure 1, we plot the volatility
estimates for the emerging markets of the sample and observe any obvious structural breaks and
shifts in the series.12 We then implement a robust procedure to detect prominent outliers in the
form of structural breaks and level shifts in conditional volatility. In particular, the robust
estimation and outlier detection procedure models the volatility series as an autoregressive and/or
moving average process and identifies the type and location of key outliers using filtered
estimates of the model parameters. The impact or the size of the outliers and their t-statistics are
obtained, which allows us to identify the most significant structural changes in volatility. This
robust change detection method is similar to those proposed by Chang et al (1988) and Tsay
(1988) and is outlined in detail in Zivot and Wang (2006).
Based on Figure 1, we observe prominent structural breaks to the volatility series for
several emerging markets. Some examples are Egypt, India, Indonesia, Morocco, Russia, and
Turkey. Also, there appear to be level shifts in volatility for several markets. A few obvious
examples are Argentina, Egypt, Jordan, Korea, and Turkey. For the majority of the markets, we
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do not identify a visible link between long memory in volatility and structural changes in the
volatility series. Three exceptions, however, are Argentina, Korea, and Turkey. Recall that all
three emerging markets documented long memory in volatility for the first but not the second
sub period. Visually, these three markets also indicate the existences of more structural breaks
and level shifts in volatility for sub period one relative to sub period two. It is important to keep
in mind that these findings are based on visual inspection alone.
For a more formal
interpretation of results, we rely on the robust outlier detection method discussed earlier.
The robust estimation procedure identified one or more significant outliers for each
market. See Table 6 for a summary of the results including the dates for the three main outliers
detected. For Greece, Indonesia, Korea, Malaysia, and Tunisia, we detect significant outliers
only for sub period one.13 Recall that, based on Table 5 results, long memory in volatility is
observed only for sub period one for these five markets. Therefore, there is some evidence that
long memory is linked with structural changes in volatility.
However, this finding is not
conclusive because we also identify nine markets that indicate long memory in volatility for sub
period one only but report significant outliers for both sub periods. In addition, Morocco and
Slovakia document structural breaks and level shifts in volatility for sub period one only but
indicate long memory for just sub period two. Consequently, we conclude that long memory and
structural changes in volatility appear to be linked for some markets, especially for several of the
Asian emerging markets. However, we do not have conclusive evidence to generalize this
finding to all the markets of the sample.
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CONCLUSION
In this paper, we investigated long memory in stock return volatility for a large group of
23 developing markets that included emerging and frontier markets from various regions of the
world. The empirical analysis was based on a fractionally integrated exponential GARCH
estimation using daily data from January 2000 to October 2007. We found significant evidence
of long memory based on the data for the entire time period. However, a sub period analysis
revealed that there is no evidence of long memory for the most recent sub period for many of the
markets in the sample. As a result, many market indices appeared to be more efficient in recent
years. We infer that the evidence of more efficient markets in recent years is likely due to the
process of market development and greater participation of competing investors. In the context
of the sub period analysis, we also explored the issue of long memory and structural changes in
volatility and found no conclusive evidence of a clear link between the two.
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Conditional heteroskedasticity in asset returns: a new approach.
Econometrica, 59(2), 347-370.
Saadi, S., Gandhi, D., and Dutta, S. (2006). Testing for nonlinearity and modeling volatility in
emerging capital markets: the case of Tunisia. International Journal of Theoretical and Applied
Finance, 9(7), 1021-1050.
Sentana, E., and Wadhwani, S. (1992). Feedback traders and stock return autocorrelations:
evidence from a century of daily data. The Economic Journal, 102(411), 415-425.
Sibbertsen, P. (2004).
Long memory in volatilities of German stock returns.
Empirical
Economics, 29(3), 477-488.
So, M. K. P. (2000).
Long-term memory in stock market volatility.
Applied Financial
Economics, 10(5), 519-524.
Tsay, R. S. (1988). Outliers, level shifts and variance changes in time series.
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Forecasting, 7(1), 1-20.
Vougas, D. V. (2004). Analysing long memory and volatility of returns in the Athens stock
exchange. Applied Financial Econometrics, 14(6), 457-460.
Wright, J. H. (1999). Long memory in emerging market stock returns. International Finance
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October 16-17, 2009
Cambridge University, UK
19
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Zivot, E., and Wang, J. (2006). Modeling financial time series with S-PLUS. New York, NY:
Springer Press.
October 16-17, 2009
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20
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Table 1: Summary statistics for daily stock returns from 1/3/2000 – 10/15/2007
Country
Mean
Median
Max
Min
Std. Dev.
Skewness
Kurtosis
p-value
Jarque-Bera
Obs
EM-Asia
China
India
Indonesia
Korea
Malaysia
Thailand
0.073
0.067
0.067
0.034
0.001
0.031
0.000
0.142
0.044
0.057
0.000
0.000
9.401
7.642
6.734
7.697
4.755
10.577
-9.256
-12.636
-10.934
-12.805
-7.093
-16.063
1.418
1.572
1.334
1.767
0.892
1.409
0.092
-0.825
-0.731
-0.581
-0.557
-0.787
8.776
8.221
8.218
7.467
10.439
14.603
0.000
0.000
0.000
0.000
0.000
0.000
2031
2031
2031
2031
1690
2031
EM-Europe, Middle East, and Africa (EMEA)
Egypt
0.096
0.000
Greece
0.000
0.000
Israel
0.043
0.015
Jordan
0.066
0.000
Morocco
0.084
0.068
Poland
0.063
0.015
Russia
0.131
0.101
Slovakia
0.085
0.000
Turkey
0.066
0.000
13.582
7.620
5.312
6.816
3.554
6.443
9.525
5.959
17.774
-7.900
-9.692
-8.959
-8.855
-6.817
-8.468
-11.071
-8.817
-19.979
1.653
1.303
1.052
1.099
0.829
1.276
2.090
1.187
2.602
0.270
-0.228
-0.588
-0.323
-0.801
-0.214
-0.434
-0.136
0.029
6.808
8.556
8.493
11.651
10.933
5.827
6.546
8.710
9.452
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
2031
2031
2031
2031
1508
2031
2031
2031
2031
EM-Latin America
Argentina
Brazil
Chile
Mexico
Peru
0.070
0.064
0.053
0.074
0.124
0.000
0.005
0.035
0.082
0.040
16.117
7.335
2.779
7.020
8.205
-11.291
-9.634
-3.854
-8.267
-7.893
2.133
1.772
0.591
1.367
1.079
0.181
-0.248
-0.431
-0.116
-0.203
8.237
4.174
6.654
6.085
10.219
0.000
0.000
0.000
0.000
0.000
2031
2031
2031
2031
2031
Frontier
Ecuador
Lithuania
Tunisia
0.062
0.086
0.041
0.000
0.063
0.000
28.932
11.866
15.023
-17.263
-13.515
-16.593
1.853
0.996
1.046
1.943
-0.711
-1.562
54.325
38.763
81.009
0.000
0.000
0.000
2031
2031
2031
Developed
US
0.003
0.002
5.573
-6.005
1.089
0.062
5.881
0.000
2031
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Table 2: Short memory and ARCH effects in returns for daily stock returns from 1/3/2000 – 10/15/2007
Q(12)
p-value
Q(12)
Q(24)
p-value
Q(24)
ARCH(12)
p-value
ARCH(12)
ARCH(24)
p-value
ARCH(24)
16
48
32
11
84
33
0.205
0.000
0.000
0.545
0.000
0.001
32
66
45
40
101
50
0.124
0.000
0.006
0.024
0.000
0.001
75
440
98
108
101
220
0.000
0.000
0.000
0.000
0.000
0.000
220
462
105
126
132
287
0.000
0.000
0.000
0.000
0.000
0.000
EM-Europe, Middle East, and Africa (EMEA)
Egypt
44
0.000
Greece
39
0.000
Israel
30
0.003
Jordan
35
0.000
Morocco
178
0.000
Poland
9
0.727
Russia
25
0.014
Slovakia
31
0.002
Turkey
18
0.112
53
60
44
52
197
23
40
38
38
0.001
0.000
0.008
0.001
0.000
0.520
0.020
0.032
0.036
242
328
139
275
169
142
231
133
322
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
254
363
176
289
174
166
253
154
343
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
EM-Latin America
Argentina
Brazil
Chile
Mexico
Peru
26
16
166
38
114
0.009
0.175
0.000
0.000
0.000
49
35
192
47
145
0.002
0.064
0.000
0.003
0.000
192
94
197
179
434
0.000
0.000
0.000
0.000
0.000
286
109
223
207
445
0.000
0.000
0.000
0.000
0.000
Frontier
Ecuador
Lithuania
Tunisia
68
73
56
0.000
0.000
0.000
79
90
76
0.000
0.000
0.000
81
282
352
0.000
0.000
0.000
88
283
351
0.000
0.000
0.000
Developed
US
20
0.065
44
0.008
294
0.000
333
0.000
Country
EM-Asia
China
India
Indonesia
Korea
Malaysia
Thailand
October 16-17, 2009
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9th Global Conference on Business & Economics
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Table 3: Long memory in return volatility for daily stock returns from 1/3/2000 – 10/15/2007
Return^2
Q(100)
p-value
Q(100)
Abs(Return)
Q(100)
p-value
Q(100)
383
1013
190
972
727
1657
0.000
0.000
0.000
0.000
0.000
0.000
764
1670
239
2046
2196
2896
0.000
0.000
0.000
0.000
0.000
0.000
EM-Europe, Middle East, and Africa (EMEA)
Egypt
429
0.000
Greece
928
0.000
Israel
349
0.000
Jordan
1613
0.000
Morocco
545
0.000
Poland
743
0.000
Russia
1004
0.000
Slovakia
331
0.000
Turkey
1542
0.000
1191
1338
465
4356
1932
895
1407
557
2438
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
EM-Latin America
Argentina
Brazil
Chile
Mexico
Peru
1735
297
612
1202
1376
0.000
0.000
0.000
0.000
0.000
1432
323
744
1556
2365
0.000
0.000
0.000
0.000
0.000
Frontier
Ecuador
Lithuania
Tunisia
202
385
714
0.000
0.000
0.000
487
380
1165
0.000
0.000
0.000
Developed
US
2559
0.000
4400
0.000
Country
EM-Asia
China
India
Indonesia
Korea
Malaysia
Thailand
Notes:
1. Return^2 Q(100) indicates the Ljung-Box Q-statistic for the squared returns series at lag 100.
2. Abs(Return) Q(100) indicates the Ljung-Box Q-statistic for the absolute returns series at lag
100.
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Table 4: FIEGARCH estimation results for daily stock returns from 1/3/2000 – 10/15/2007
Country
GARCH
ARCH
Leverage
Fraction
China
0.267
(0.025)
0.180
(0.000)
-0.024
(0.002)
0.703
(0.000)
India
0.681
(0.000)
0.299
(0.000)
-0.138
(0.000)
0.304
(0.000)
Indonesia
0.694
(0.000)
0.201
(0.000)
-0.146
(0.000)
0.193
(0.001)
Korea
0.681
(0.000)
0.132
(0.000)
-0.088
(0.000)
0.478
(0.000)
Malaysia
0.370
(0.006)
0.157
(0.000)
-0.056
(0.000)
0.601
(0.000)
Thailand
0.761
(0.000)
0.135
(0.000)
-0.085
(0.000)
0.404
(0.000)
EM-Asia
EM-Europe, Middle East, and Africa (EMEA)
Egypt
0.166
(0.021)
0.342
(0.000)
0.023
(0.061)
0.541
(0.000)
Greece
0.663
(0.000)
0.180
(0.000)
-0.088
(0.000)
0.442
(0.000)
Israel
0.824
(0.000)
0.165
(0.000)
-0.090
(0.000)
0.231
(0.001)
Jordan
0.842
(0.000)
0.056
(0.000)
0.027
(0.000)
0.597
(0.000)
Morocco
0.391
(0.000)
0.457
(0.000)
-0.018
(0.134)
0.546
(0.000)
Poland
0.914
(0.000)
0.078
(0.000)
-0.006
(0.131)
0.385
(0.001)
Russia
0.719
(0.000)
0.201
(0.000)
-0.078
(0.000)
0.303
(0.000)
Slovakia
0.736
(0.000)
0.201
(0.000)
0.010
(0.151)
0.305
(0.000)
Turkey
0.496
(0.000)
0.179
(0.000)
-0.055
(0.000)
0.554
(0.000)
October 16-17, 2009
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24
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Table 4 (continued): FIEGARCH estimation results for daily stock returns from
1/3/2000 – 10/15/2007
Country
GARCH
ARCH
Leverage
Fraction
Argentina
0.480
(0.000)
0.153
(0.000)
-0.043
(0.000)
0.580
(0.000)
Brazil
0.519
(0.000)
0.070
(0.001)
-0.158
(0.000)
0.392
(0.000)
Chile
0.288
(0.013)
0.314
(0.000)
-0.037
(0.008)
0.615
(0.000)
Mexico
0.604
(0.000)
0.166
(0.000)
-0.136
(0.000)
0.466
(0.000)
Peru
0.578
(0.000)
0.367
(0.000)
-0.033
(0.007)
0.378
(0.000)
Ecuador
0.531
(0.000)
0.346
(0.000)
-0.051
(0.000)
0.355
(0.000)
Lithuania
0.622
(0.000)
0.275
(0.000)
-0.002
(0.409)
0.364
(0.000)
Tunisia
0.297
(0.000)
0.371
(0.000)
0.034
(0.000)
0.573
(0.000)
0.343
(0.002)
0.074
(0.000)
-0.202
(0.000)
0.632
(0.000)
EM-Latin America
Frontier
Developed
United States
Note: For all estimations, p-values are given in parenthesis. The coefficient estimates for the
conditional mean equation with the relevant ARMA terms are available upon request.
October 16-17, 2009
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25
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Table 5: FIEGARCH sub period estimation results for the leverage and fraction estimates
Country
Sub sample 1 (1/3/2000-12/31/2003)
Leverage
Fraction
Sub sample 2 (1/1/2004-10/15/2007)
Leverage
Fraction
EM-Asia
China
-0.090
(0.000)
0.631
(0.000)
0.011
(0.074)
0.097
(0.250)
India
-0.113
(0.000)
0.480
(0.000)
-0.257
(0.000)
0.179
(0.008)
Indonesia
-0.116
(0.000)
0.199
(0.084)
-0.199
(0.000)
0.000
(0.500)
Korea
-0.106
(0.000)
0.590
(0.000)
-0.200
(0.000)
0.000
(0.500)
Malaysia
-0.092
(0.000)
0.538
(0.000)
-0.010
(0.149)
0.255
(0.116)
Thailand
-0.063
(0.000)
0.499
(0.000)
-0.124
(0.000)
0.000
(0.500)
EM-Europe, Middle East, and Africa (EMEA)
Egypt
0.054
(0.006)
0.462
(0.000)
-0.041
(0.048)
0.616
(0.000)
Greece
-0.087
(0.000)
0.427
(0.000)
-0.128
(0.000)
0.000
(0.500)
Israel
-0.083
(0.000)
0.042
(0.415)
-0.168
(0.000)
0.000
(0.500)
Jordan
0.031
(0.000)
0.528
(0.000)
0.025
(0.000)
0.623
(0.000)
Morocco
-0.021
(0.233)
0.165
(0.179)
-0.006
(0.409)
0.636
(0.000)
Poland
-0.015
(0.142)
0.020
(0.462)
0.001
(0.421)
0.576
(0.020)
Russia
-0.068
(0.000)
0.384
(0.000)
-0.091
(0.000)
0.000
(0.500)
Slovakia
0.017
(0.142)
0.000
(0.500)
0.011
(0.270)
0.247
(0.000)
Turkey
-0.052
(0.001)
0.364
(0.000)
-0.111
(0.000)
0.000
(0.500)
October 16-17, 2009
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26
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Table 5 (continued): FIEGARCH sub period estimation results for the leverage and fraction
estimates
Country
Sub sample 1 (1/3/2000-12/31/2003)
Leverage
Fraction
Sub sample 2 (1/1/2004-10/15/2007)
Leverage
Fraction
EM-Latin America
Argentina
-0.030
(0.015)
0.767
(0.000)
-0.117
(0.000)
0.042
(0.394)
Brazil
-0.160
(0.000)
0.650
(0.000)
-0.291
(0.000)
0.372
(0.000)
Chile
0.019
(0.207)
0.697
(0.000)
-0.098
(0.000)
0.012
(0.463)
Mexico
-0.128
(0.000)
0.565
(0.000)
-0.167
(0.000)
0.000
(0.500)
Peru
-0.003
(0.447)
0.535
(0.000)
-0.057
(0.008)
0.226
(0.011)
Ecuador
-0.051
(0.000)
0.000
(0.500)
-0.050
(0.000)
0.566
(0.000)
Lithuania
-0.025
(0.176)
0.271
(0.008)
-0.020
(0.109)
0.000
(0.500)
Tunisia
0.016
(0.142)
0.534
(0.000)
0.040
(0.026)
0.000
(0.500)
-0.216
(0.000)
0.720
(0.000)
-0.171
(0.000)
0.000
(0.500)
Frontier
Developed
United States
Note: For all estimations, p-values are given in parenthesis. The coefficient estimates for the
conditional mean equation with the relevant ARMA terms are available upon request. The
complete estimation results for the conditional variance equation are also available upon request.
October 16-17, 2009
Cambridge University, UK
27
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Table 6: Outliers (structural breaks and level shifts) detected in the volatility series using a robust detection procedure
Country
Number of outliers detected
sub 1
sub 2
EM-Asia
China
India
Indonesia
Korea
Malaysia
Thailand
6
2
1
4
4
10
7
2
0
0
0
7
Type of outlier
Structual break
Level shift
sub 1
sub 2
sub 1
sub 2
Dates of outliers
5
0
0
3
3
7
6
2
0
0
0
5
1
2
1
1
1
3
1
0
0
0
0
2
2/28/2007, 2/15/2000, 6/5/2007
5/18/2004, 5/19/2006, 4/5/2000
10/15/2002
4/18/2000, 9/13/2001, 9/19/2000
4/18/2000, 9/18/2001, 4/5/2001
3/14/2000, 11/21/2000, 7/30/2007
EM-Europe, Middle East, and Africa (EMEA)
Egypt
4
5
Greece
3
0
Israel
2
0
Jordan
2
4
Morocco
1
0
Poland
3
2
Russia
8
9
Slovakia
3
0
Turkey
7
1
3
2
1
0
1
1
7
0
6
5
0
0
0
0
0
8
0
1
1
1
1
2
0
2
1
3
1
0
0
0
4
0
2
1
0
0
4/27/2005, 2/3/2003, 9/23/2005
4/18/2000, 9/13/2001, 3/15/2000
10/13/2000, 4/17/2000
9/19/2001, 1/4/2005, 7/26/2005
1/3/2003
4/18/2000, 8/17/2007, 1/6/2000
12/1/2000, 12/18/2000, 1/10/2007
12/4/2001, 3/14/2000, 12/18/2002
2/22/2001, 3/4/2003, 7/9/2001
EM-Latin America
Argentina
Brazil
Chile
Mexico
Peru
11
3
6
10
1
3
1
6
4
1
7
0
5
9
0
3
1
5
3
1
4
3
1
1
1
0
0
1
1
0
10/30/2001, 2/28/2007, 3/5/2002
2/28/2007, 9/12/2001, 1/5/2000
5/23/2007, 6/14/2006, 8/20/2007
4/17/2000, 1/5/2000, 2/28/2007
6/1/2007, 9/19/2000
Frontier
Ecuador
Lithuania
Tunisia
5
1
3
3
1
0
4
1
2
3
1
0
1
0
1
0
0
0
8/23/2005, 2/22/2002, 6/13/2003
12/5/2000, 4/26/2005
8/1/2001, 6/4/2001, 1/13/2000
Developed
US
3
1
1
1
2
0
4/17/2000, 3/13/2001, 2/28/2007
Note: The last column gives (up to) the three most significant outliers detected based on the impact of the outlier.
October 16-17, 2009
Cambridge University, UK
28
9th Global Conference on Business & Economics
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Figure 1: Fractionally integrated conditional volatility estimates for the emerging markets from 1/3/2000 to 10/15/2007
8
7
8
Argentina
7
8
Brazil
7
8
China
7
8
Chile
7
6
6
6
6
6
5
5
5
5
5
4
4
4
4
4
3
3
3
3
3
2
2
2
2
2
1
1
1
1
1
0
0
0
0
2000 2001 2002 2003 2004 2005 2006 2007
8
7
2000 2001 2002 2003 2004 2005 2006 2007
8
Greece
7
2000 2001 2002 2003 2004 2005 2006 2007
8
India
7
0
2000 2001 2002 2003 2004 2005 2006 2007
8
Indonesia
7
2000 2001 2002 2003 2004 2005 2006 2007
8
Israel
7
6
6
6
6
6
5
5
5
5
5
4
4
4
4
4
3
3
3
3
3
2
2
2
2
2
1
1
1
1
0
0
2000 2001 2002 2003 2004 2005 2006 2007
8
7
0
2000 2001 2002 2003 2004 2005 2006 2007
8
Korea
7
8
Malaysia
7
0
2000 2001 2002 2003 2004 2005 2006 2007
8
Mexico
7
2000 2001 2002 2003 2004 2005 2006 2007
8
Morocco
7
6
6
6
6
6
5
5
5
5
5
4
4
4
4
4
3
3
3
3
3
2
2
2
2
2
1
1
1
1
1
0
0
0
0
2000 2001 2002 2003 2004 2005 2006 2007
8
7
2000 2001 2002 2003 2004 2005 2006 2007
8
Poland
7
2000 2001 2002 2003 2004 2005 2006 2007
8
Russia
7
7
2000 2001 2002 2003 2004 2005 2006 2007
8
Thailand
7
6
6
6
6
6
5
5
5
5
5
4
4
4
4
4
3
3
3
3
3
2
2
2
2
2
1
1
1
1
1
0
0
0
0
2000 2001 2002 2003 2004 2005 2006 2007
2000 2001 2002 2003 2004 2005 2006 2007
2000 2001 2002 2003 2004 2005 2006 2007
Note: The shaded area indicates sub period two.
October 16-17, 2009
Cambridge University, UK
29
Peru
0
2000 2001 2002 2003 2004 2005 2006 2007
8
Slovakia
Jordan
1
0
2000 2001 2002 2003 2004 2005 2006 2007
Egypt
T urkey
0
2000 2001 2002 2003 2004 2005 2006 2007
2000 2001 2002 2003 2004 2005 2006 2007
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Endnotes
1
Many have also examined long memory in the level of stock returns. Some examples are Assaf (2006), Barkoulas et al (2000), Chen
et al (2001), Christodoulou-Volos and Siokis (2006), Vougas (2004), and Wright (1999).
2
In forecasting long memory models, Calvet and Fisher (2001) have recently introduced a Markov-switching multi-fractal model that
can account for different degrees of long term dependence of financial data. This model has shown the potential to produce better
forecasts than some of the fractional integration models.
3
Three exceptions are Chung et al (2000) who use 43 individual stocks listed in the Taiwan Stock Exchange in their data analysis,
Sibbertsen (2004) who tests for long memory in volatility for seven individual German stocks, and So (2000) who focuses part of his
study on the 30 constituent stocks of the Dow Jones Industrial Average (DJIA) index.
4
Wright (2002) finds that estimators generally exhibit some downward bias if data is conditionally Gaussian and that this downward
bias is greatly increased if squared returns are selected as the volatility measure.
5
Chung et al (2000) also suggest aggregation as another possible cause of spurious long memory. However, they find evidence of
long memory in volatility for many of the 43 individual stocks studied in addition to the aggregate stock index studied for Taiwan.
6
Given the iterative nature of the conditional volatility specification in equation (4), a maximum likelihood procedure is used to
obtain the model coefficients.
October 16-17, 2009
Cambridge University, UK
30
9th Global Conference on Business & Economics
7
ISBN : 978-0-9742114-2-7
Data for Malaysia is available only from January 3, 2000 to June 24, 2006. For Morocco, data is available only from April 1, 2002
to October 15, 2007.
8
The returns series are all stationary as confirmed by relevant unit root tests based on the Augmented Dickey Fuller (ADF) test. The
ADF test results are available upon request.
9
In the existing literature the squared, log-squared, modified log-squared, and absolute returns are often used as proxies for volatility
of returns.
10
Regression estimates for the GARCH and ARCH effects are available upon request.
11
A closer look at the Israel stock index returns is warranted given evidence of long memory for the entire time period but not for the
two individual sub periods.
12
The graphs for the frontier markets and the U.S. are not presented in order to conserve space, and are available upon request.
13
The emerging market of Turkey may be considered a part of this group too because there is only one observed outlier for sub period
two and not one of the top three outliers in Turkey belongs to sub period two.
October 16-17, 2009
Cambridge University, UK
31